Parallel computer systems are easier to build than to use effectively. One goal of this research is to implement, measure, and analyze the performance of a wide variety of algorithms on current generation parallel computers. The PIs wish to gain experience with, and insight into, the practical problems arising in the efficient use of such machines to solve realistic computational problems using a wide variety of algorithm design techniques. A second goal is to use experience gained from this implementation effort to develop refined theoretical models of such computers. The models that are most popular in the theoretical community are felt to be oversimplified. It is hope that new models can be formulated that remain simple, yet better focus attention on the key performance issues relevant to these parallel machines.